Remark 1 Assume that is invariant under a transitive group of permutations of . Then

This can be viewed as an isoperimetric inequality.

1.6. Proof of KKL theorem

It uses some discrete harmonic analysis. Use the notation from the courses by Kindler or Raghavendra. Every function (Fourier-Walsh) decomposes as

where

Parseval’s identity is

Denote by

In general,

thus

Thus influences read

Assume (by contradiction) that for all , . Then all have small support. Heisenberg’s uncertainty principle suggests that this implies that spectrum is high. Bonami-Beckner inequality makes this quantitative. On the other hand,

If spectrum is high, This is much larger than .

2. Application to statistical physics

2.1. Russo’s formula

Proposition 5 Let be monotone. Use the -biased distribution on the hypercube. Then the derivative of at is the total influence .

The KKL theorem implies the following lower bound (BKKKL), for all , with a -independant constant ,

For symmetric functions,

So the slope is big, this is an instance of a sharp threshold result.

One can view this as a generalization of Kolmogorov’s -law.

Theorem 6 Let be a monotone set. Then is either or .

2.2. Bond percolation

Kesten’s proof that for bond percolation on . He knew that is balanced at , he proved by hand that influences were large, so just above connections across rectangles of aspect ratio have high probability.

2.3. First passage percolation

Study balls in random metrics on . One knows that a limit shape exists. KKL has been used to show that fluctuations around that shape are small.

2.4. Dynamical percolation

See Gabor Pete’s talk on friday.

3. Randomized algorithms

3.1. Revealment

Given a Boolean function , one wants an algorithm that computes with high probability, using few queries, i.e. minimizing the revealment

Theorem 7 (Schramm, Steif) For every Boolean function , for all , for all algorithms ,

This means that small revealment gives a strong control on the Fourier coefficients of .

There is an algorithm with revealment for the connection function: launch a sample connection, i.e. travel through the domain querying at each step the colour of the encountered hex. This requires queries.

There is a conjecturally better algorithm.

3.2. Queried sets

Let denote the set of variables queried by . It is easy to show that

In fact, R. O’Donnell and R. Servedio could prove that

This is very useful, since it works in the opposite direction as KKL, and gives lower bounds on revealment, since

For instance, for the connection Boolean function, is bounded from above by the typical length of an curve, i.e. . Compare to the lower bound provided by Peres, Schramm et al.

3.3. Witness revealment

Theorem 8 (Schramm, Steif) For every Boolean function , if there exists an algorithm with small revealment, then is noise sensitive.

Define a witness as a set of coordinates that suffice to prove that (e.g. a connection path for the connection Boolean function). The witness revealment of is the infimum over all randomized witnesses which compute of